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NBS MONOGRAPH 102

Effects of Finite Lattice

Heat Capacity on

Spin-Lattice Relaxation

Theory and Numerical Analysis

National Bureau of Standards JUL 2 8 1S72 17120*4

QC

1 00

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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Effects of Finite Lattice Heat Capacity

on Spin Lattice Relaxation

Theory and Numerical Analysis

Robert L. Peterson

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

National Bureau of Standards Monograph 102

Issued August 1, 1967

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington. D.C. 20402 -Price 25 cents
«
Contents

Page

1. Introduction 1

2. Theory 2

3. Spin relaxation for tightly coupled vibrational modes 6

4. Spin relaxation for uncoupled vibrational modes 8

5. Spin relaxation to the resonant phonons coupled to a bath 18

6. Discussion and summary 20

7. References 21

ill

Effects of Finite Lattice Heat Capacity on

Spin-Lattice Relaxation

Theory and Numerical Analysis

Robert L. Peterson

The transient magnetic behavior of a paramagnetic substance, after an initial disturbance, is considered theoretically for a variety of situations in which the lattice temperature rises as a result of energy flow from the magnetic (electron spin) system. The relaxation mechanisms considered are the direct process, involving interaction between spins and the resonant-phonon modes, and a T9 Raman process, involving the remaining phonon modes. It is first assumed that the resonant and

remaining modes are strongly coupled to each other, and that the helium bath has been removed

(helium temperatures are assumed). The resulting transient behavior typically does not differ much

from exponential relaxation (the rate increases somewhat during the relaxation), but the difference should be experimentally observable, particularly if the resonance line is inverted initially. Next, the opposite extreme is considered — that in which the resonant and remaining modes are

totally uncoupled; the helium bath is again assumed to have been removed. Two dramatic effects can then occur, depending upon initial conditions. One is the rapid decay to saturation from initial inversion, due to an avalanching creation of resonant phonons by the spins. The second is the pro- nounced inhibition of the subsequent decay, with the spins remaining near saturation. This decay

rate is typically very much slower than the Raman rate, and is due to the flow of the resonant-phonon

energy back into the spin system as the spin energy flows into the remaining lattice. Eventually the

rate increases to the Raman rate characteristic of the final lattice temperature. Because of inelastic phonon scattering at crystal boundaries, which couples the phonon modes together, this effect of inhibited decay may be difficult to observe.

Finally, we consider the case of spins coupled only to the resonant phonons, which in turn are coupled to a constant temperature bath, whether this be the helium or the remaining modes. Recent phonon avalanche experiments are discussed in this context. It is pointed out that such experiments,

performed in the absence of a bath, may provide a reliable measurement of inelastic phonon-boundary

scattering.

Key Words: Spin-lattice relaxation, phonon bottleneck, phonon-boundary scattering, spin-phonon

interactions.

1 . Introduction

It is our purpose in this Monograph to present

many additional graphs of the transient magnetic

behavior of a paramagnetic system, to supplement

a paper published earlier [Peterson, 1965a]. These

graphs show the possible behavior of the magnetization of a paramagnet under conditions in which

the lattice containing the magnetic moments can

heat up as a result of the magnetic relaxation.

We here consider a broader range of parameter

values than in the earlier publication in order to demonstrate the types of transient behavior which

cari occur under various conditions. With two

exceptions, the parameter values used do not cor- respond to specific, known materials. They are chosen, rather, to illustrate what can occur in situ- ations which are physically realizable, although in some cases it may indeed be difficult to approximate

closely the models used here. In some materials,

there will be more relaxation mechanisms operative

than we have considered here, some of which may

be as important or more important than those which we take into account. For such materials, the

results given here will provide an estimate of the

influence of those relaxation mechanisms which we

have considered. In addition to considering broader ranges of

parameter values, we here make additional application of the theory used in the earlier work [Peter- son, 1965a]. We also extend the theory and apply

it to situations not considered previously.

The theory used in this Monograph is developed

in section 2. In sections 3 and 4, the conventionally

used "bath," typically liquid helium, is assumed to have been removed, or otherwise made ineffective. Heat conduction into or through the crystal support

is also assumed to be negligible during the relaxation. Ruby, Benoit, and Jeffries [1962] have

reported an experiment using adiabatic demagnetization techniques in which these conditions would seem to be satisfied. Thus, by assumption, energy

cannot flow out of the lattice subsequent to a dis- turbance in the system of magnetic moments. It

is this absence of an effective bath which allows

the interesting, atypical transient behavior of the

magnetization which is the main subject of this Monograph.

1
In sections 3 and 4, the magnetic moments, hereafter called spins, constitute one system; the lattice vibrational modes which couple directly to the spins,

and called the resonant modes, constitute a second

system; and the remaining vibrational modes, which

couple to the spins via Raman processes, constitute

the third system. In section 3 it is assumed that the resonant and remaining vibrational modes are

tightly coupled to one another as compared with

their coupling to the spins. This is the case whenever it is observed experimentally that the total spin relaxation rate is the simple sum of the two

separate relaxation rates. Section 4 treats the

opposite extreme: We assume that the resonant modes are completely uncoupled from the remaining

vibrational modes. This is realized physically

whenever the spin relaxation times are small compared to the characteristic coupling times of the

lattice, i.e., the phonon-phonon and phononboundary collision times. At helium temperatures,

phonon mean-free-paths as limited by phononphonon interactions are usually much greater than

the dimensions of the sample. The coupling between modes at the temperatures of interest here

is therefore indirect, and results from inelastic scattering of the phonons from boundaries [Ziman,

1960; Casimir, 1938]. The effects of "moderate"

coupling between the vibrational modes lie between

the extreme cases treated in sections 3 and 4, and

are readily estimated for various effective phononphonon relaxation times. In section 5 we re-introduce the constant-temperature bath, allowing it to interact with the lattice vibrations. We consider only the direct coupling

of the spins to the resonant modes, assuming the

coupling of the spins to the remaining modes to be

unimportant, often an excellent assumption. Our

reason for investigating this situation is that it is the dominant coupling scheme in the recent "phonon

avalanche" experiments of Brya and Wagner [1965,

1966] and Shiren [1966]. Although we had pre- dicted such behavior in general terms in our earlier paper [Peterson, 1965a], we did not consider this specific case there, and wish to examine it in some

detail here. Section 6 summarizes the results, and gives some

additional discussion. It is pointed out that spin

relaxation in the absence of a bath should provide a useful method for studying the effective phononphonon interactions, whether these be direct or

indirect. Figure 1 is a block diagram showing the possible

combinations of energy flow considered in this paper, and will be referred to in subsequent sections.

Most computations were made with the aid of a

digital computer. Where analytical reduction was

possible, comparison was made to verify the accu- racy of the computer results. Where analytical

reduction was not possible, the "mesh" size (Af) was

varied to ensure the accuracy of the computer

results.

Figure 1. Block diagram of the possible channels of energy

flow considered in this Monograph.

2. Theory

In this section, we develop the theory underlying

the rate equations used in subsequent sections, and

make an explicit calculation of the relaxation time

characterizing the direct process. Because these

rate equations are essentially those which are commonly used, except that here all temperatures

are allowed to be time-dependent, the reader

familiar with these equations may skip this section

without impairment to an understanding of the

remainder of the paper. Our reason for wanting

to develop the underlying theory here is that it does

not seem to have been treated carefully elsewhere,

except briefly in our earlier paper [Peterson, 1965a].

We also wish to emphasize that a single, unified

development can treat a variety of situations, more

than just those considered in this paper. The principal hypothesis is that each system is separately in internal equilibrium, with a well- defined temperature. The temperatures of the

various systems are not necessarily equal, and the

couplings between the systems bring these temperatures to a common value. It is usually necessary

to assume that these couplings are sufficiently weak

compared to the interactions internal to the systems,

that the temperatures remain well defined while

energy flows between the systems. An important

exception to this is the two-level system. Another

example is discussed later in this section. In these cases the concept of temperature does not need to be introduced, although it may be convenient and

intuitively helpful.

The time rate of change of the occupation probabilities of an isolated system, which in our case

consists of the set of weakly coupled systems, is given approximately by the Pauli master equation, an equation well suited for the study of many relax- ation phenomena [Peterson, 1965a and 1965b].

2
This equation is 4 WW= 2 ww lw\PW)-<WWt\ (i) dt %

Here (t//|P|i|/) is the probabihty that the group of

systems is in state which is an eigenstate of the sum of the Hamiltonians of the individual systems

(excluding the coupling between the systems, which

constitutes the perturbation V). is the transition probabihty per unit time, which in lowest

order isWw = (27r/7*) | (^W)

|

2 8{£* -£*}, (2)

where b{E^ — E^} is the delta function expressing

conservation of energy in the group of systems.

The state for our purposes here may be written

as a product of the states 1 of the separate systems:

|«Jr) = |aj, a2 , . . .)= |ai)|a2 ) . . . ,

where |a,) is the energy eigenstate for system i, with eigenvalue Ei(aii). The average energy of system i at time t is (^>t = £ a< • • • l Pl ai < • > =

a,a2 . . - = ^Ei (ai )(ai \p i \ai ), (3)

where

{m\pi\cLd= £ (a,, . . . \P\au •)• (4)

The sum in eq (4) excludes system i. <a,|pi jet,) is just the probabihty of finding system i in state \oa).

Multiplying eq (1) by Ei(ca) and summing on all states one obtains ^t m= 2 w**- [Edm)

W\PW). (5)

where ) = |a,-' ) |a2

' ) .... The right side of

eq (5) is obtained with use of the fact of "microscopic

reversibility," — W$$, which follows from eq (2). The right side of eq (5) is of order V

2

. The probabihty (t|/' in eq (5) can therefore be factored

as <i/r'|P|*'>=n<af|p<|ai>, < 6>

i

since the correction to eq (6) will be at least of order

V, and will not affect the validity of eq (5) to order V

2

.

1 This factorization is not possible in the longitudinal spin-spin relaxation problem. See [Peterson, 1965b] and (Philippot. 1964|.

Now V is the sum of the pairwise couplings Vmn between all systems. Since Vmn contains operators

referring only to systems m and n, the matrix ele- ments (ai, . . . afc , . . . \Vmn \oii', . . . <Xk , . . .) of Vmn vanish unless \ak) = \otk') for all k except

m and n. Therefore, because of the factor Ei(ai) — Ei(ai') on the right in eq (5), the Vmn terms in cannot contribute unless either m — i or n~i.

That is, the rate of change of the energy of system i is affected directly only by those systems coupled

to system i, which is an intuitive result. The

remaining terms in W^< are of the form {y\i\V

m \\\i'

)

(if/'\ Vin | The latter vanish unless \otk) —

for all k except i. The argument of the delta

functions multiplying these cross product terms

thus reduces to Ei(cti) —Ei(a.i) . But this argu- ment is also the factor appearing in eq (5), and so the cross product terms cannot contribute. Thus

eq (5) can be written

d(^i )ldt = ^Aij , (7)

j where

Aij= 2 W(am' , am' ) [Ei(aci) —Ei(ai')]

<!>. V

n

[ {ak '\pk\ak ') Yl' oV,', (8)

fc=l { and

W(am\ ajaj') = {27THi)\(aia} \Vij \ai 'aj ')\ 2

S{Ei(oti) +Ej(aj) -Edccj') -JEj(a[)}. (9) The product Y[ m e 9 (8) excludes systems i and j.

i Since ^ (ak \pk \ak ) = l,

a

k Ajj can be written

Aa= 2 2 W{am\ aja/) [Et (cti) a i- ai <*j a

j

—Ei(ai')] (ai'\pi\ai') (a/lp^a/) . (10)

Note that eqs (9) and (10) require

Ai^-Aji. (11)

Thus, if eq (7) is summed over all systems i, eq

(11) shows that the total energy (excluding the

weak couplings) is conserved. In passing from eq (1) to eq (7), no approximations

have been made, other than that of eq (5), which

does not affect the validity of eq (7) to order V

2

, which is the order of eq (1). Now we introduce
the hypothesis of thermodynamic equilibrium within

each subsystem, with time-dependent temperatures.

This idea was apparently first used by Casimir and

Du Pre [1938], and has been successfuly applied

in many instances since. The left side oi eq (7) can then be written

d{Wj)

dt dpi dt

Cj dpi

kfli

2 dt

(12)

where C, is the heat capacity of system i, and

fii(t) = [kTi(t)]- 1

, where k is Boltzmann's constant, and Ti(t) is the temperature. In thermal

equilibrium, (a,|p,|aii) will have the form

(ai\pi\ai)

exp [-fr(Q£i(ai)]

£exp [-#(t)&(«i')j

(13)

Substitution of eqs (12) and (13) into (7) then gives a set of coupled first-order, nonlinear differential equations in the variables Bu

Specializing to a group of three systems, one can

write, using eqs (7) and (12), —dp, /3 2 -/3i fo-jS t, ... .

1 (14a) dt T12 T13 *= £-& fil2+ &z&. (14b)

dt Ti2 T23 dp3 _ P1 ~P3 P2 ~P3 t(i3-\ ng23, (14c)

dt Ti3 T23 where

1 =

kp? Ajj =

dp?

T« CiiPi-pj)'

,J CjP?

(15)

The "relaxation times" Ty and ratios Rij of eqs

(14) and (15) are of course time dependent, becoming

constant asymptotically in time. (Ay becomes

proportional to Pi — Pj in this limit, as may be seen from eq (18) below.) As we have shown earlier [Peterson, 1965b and 1965a], the asymptotic value

of Ty, when system i is a spin system, is the lowtemperature generalization of a formula derived by

Gorter [1947] and developed further by Hebel and

Slichter [Hebel, 1963].

If system i is a spin system s, one may conveniently, and often with negligible loss of accuracy, use a high temperature approximation throughout,

as we now show. (If one is interested only in the

asymptotic relaxation times of the set of systems,

this approximation is readily avoided [Peterson,

1965b].) We assume that the spins, Ns in number,

are independent of one another, the usual assumption in spin-lattice relaxation. The spin density

matrix can then be factored into a product of density matrices, one for each spin. Further, W(asas \

otj(Xj') decomposes into a sum of transition probabilities, one for each spin, by arguments similar

to those preceding eq (7). Taking all spins to be

identical, one readily finds that

Asj = Ns £ ^ Wimm^aja/j (Em -Em>)

X(m'\ps \m') (a'jlpjla'j) (16a)

o m,m' ctj,ctj'

[(m'\p s \m') (aj\pj\atj) — (m\ps \m) (aj\pj\aj)],

(16b)

where Em is the energy and (m\ps \m) is the spin

density matrix element of a single spin in state

| to) , and

W{mm', ajcx'j) = (2TrlH)\{maj \Vsj\m'a'j )\

2

8{Em -Em, + Ej (aj)- Ejia'jfi

.

(17)

Equation (16b) is just the symmetrized form of ASj. Substituting eq (13) into (16b), and noting that (m\ps \m) (a.j\pj\a.j) , when multiplied by the energy

conserving delta function of eq (17), can be written

(m'\ps \m') (a'j\pj\a'j) exp [(&•-&) (Em -Em>)],

one obtains

Asj = Ns ^ (Em-Em) [1- exp (fij-Ps) (Em-Em)]

(m'\ps \m r ) Wj (m' , m) , (18)

where the summation is over all pairs of spin states, and

Wj(m', m) = ^ W(mm', aja/) (aj|pj|a}). (19)

[The quantity wj(m' , m) is the transition probability per unit time from state \m') to state \m),

taking into account the influence of system j. By

multiplying and dividing the right side of eq (19) by ( aj\Pj\<Xj), and using eqs (13) and (17), one

can obtain

wj(m', m) = ^ W{mm\ aja/) (aj\pj\aj)

araj'

exp [Pj(Em>— Em)] =Wj(m, m') exp (20)

where wj(m, m') is defined by eq (19) with m and

m' interchanged. It is just the transition probability per unit time from \m) to \m'). Equation (20)

is the "principal of detailed balance," which ensures

that our rate equations are governed by both stimu4
lated and spontaneous processes, as they must be

in a proper quantum mechanical treatment.]

The high-temperature approximation, referred

to earlier, and often used in spin resonance and

relaxation problems, consists of expanding the

(exponential function of eq (18) and retaining only

the term linear in (/3j — /3S) {Em'~Em). This quantity is usually much less than unity in magnitude,

, although for our purposes it is sometimes initially

about unity. Suppose, for example, that one has

initial spin inversion (the worst case from the

standpoint of the high-temperature approximation),

with Tj(0) =- n(0) = 1 °K. Then

_ Ifr— &) (Em>-Em)\ «0.1 v,

where v is in GHz (10 9 cps). At Z-band (^ = 10)

this quantity is about unity. As the relaxation progresses, it of course becomes smaller. We

show in figure 2 the transient magnetization after

initial spin inversion, calculated with and without

the high-temperature approximation. A £-band

level splitting (^ — 25, or A/A ~ 1 °K) is used, with

initial lattice temperatures of 1 °K and 2 °K. Considered in figure 2 is the case of the spins interacting

only with the resonant phonons (channels b, c. d,

and e in figure 1 are closed). This case is treated

in section 5. The high-temperature approximation

(equivalent to A/A = 0) is seen to give a slightly faster spin relaxation rate initially than the equations without this approximation give. The approximation therefore underestimates slightly the delay

time before the rapid decay (the "phonon ava- lanche") characteristic of initial spin inversion,

begins. The differences are sufficiently small that one is well justified in using the much simpler hightemperature version of the rate equations for fre- quencies at K-band or lower, and initial lattice temperatures of about 1 °K or higher.

0 100 200 U/xsec

)

FIGURE 2. Transient magnetization after initial spin inversion, for two initial lattice temperatures Td(0), as calculated exactly

for K-band splitting (A/k = 1 °K), and in the high-temperature

approximation (A/k = 0). Parameters used are o= 101 <ir%* and aj= 1 sec" 1 deg-1

. See text for definitions.

There are two other points to consider in the

high-temperature approximation. One concerns

the spin heat capacity, which contains terms such

as exp(— /3SA) where A= \Em> — Em \. For consistency one must write exp(— /3SA) = 1 — /3SA. The spin heat capacity is then

Cs = \s(S+\)Ns A

2 A/3S2 (21)

for Ns independent particles of spin S and energylevel separation A. (The concept of a single spin

temperature usually is valid only if there is but one

pair of spin levels, or if the spin levels are equally

spaced. We explicitly consider only the former,

using an effective spin of one-half.)

The other point concerns the high-temperature

expression for Ay where neither i nor j represents

the spin system. Although in this paper we do not have explicit need for such an expression, we point out that the high-temperature approximation may

be applied just as with the spin system. The reason

is that, in our work, either i or j is always the system

of resonant phonon modes, which by definition have

energies equal to the energy level splittings in the

spin system.

The high-temperature expression for TSj for a

collection of effective spin one-half particles is therefore, by eqs (15), (18), and (21).

Tj/ = 2wj(m', m). (22)

Here, Wj(m' , m) may be used interchangeably with

Wj(mm') because of the high-temperature approximations already used. Because system j is a phonon system, and because eq (22) is derived from

the lowest-order transition probability expression —

which means that only one phonon is created or destroyed when a spin flips — the quantity t~\

is the direct relaxation rate, hereafter denoted by t^1

. For Wj(m' , m) given by eq (19), it is well known,

and not too difficult to show, that fori?,,,—

£' m'=A>0,

and system j a boson field (phonons or photons),

Wj{m' , m) =A nj(A),

where A is the Einstein coefficient for spontaneous

emission, andn)(A) is the Planck factor, [efy A — 1] _1

, at temperature 7j(r), giving the number of thermal

phonons per mode at energy A. The direct relax- ation rate is therefore finally given by

t^1= (2Ak/A)Tj(t) , (23)

in the high-temperature approximation. We remark

that this result can be achieved more rapidly by

beginning with intuitive rate equations for the spin

occupation numbers and phonon occupation numbers. It has been obtained in this way for constant

Tj (constant phonon occupation numbers) by many

workers. The generalization to a time-dependent

5
resonant-phonon temperature has not been used

explicitly in the literature, with the exception of our own earlier work [Peterson, 1965a]. The notion

of a steady-state resonant-phonon temperature has

been used, however [Van Vleck, 1941; Orbach,

1961; Faughnan and Strandberg, 1961; Scott and

Jeffries, 1962]. Actually, the resonant-phonontemperature language is just an alternate language

to that of the phonon occupation numbers, the two

concepts being tied together by the Planck factor. The temperature concept is quite convenient to use,

as is evidenced by the manner of its appearance in the equations in the following sections, and in many

instances is a more intuitively meaningful measure

of the degree of phonon excitation than is the

phonon number.

The Raman process is a higher-order process,

involving any two phonons whose energy difference

is equal to that of the spin-flip energy. This process

thus makes use of the entire phonon spectrum, with

the exception of the resonant phonons. Our use

of a single temperature to describe these phonons

is now more than a convenient change in language —

it is a physical assumption requiring that the pho- nons interact with one another either directly or

indirectly, sufficiently rapidly compared to the spinphonon interactions that they remain thermalized.

The Raman relaxation rate r^ 1 is considerably more difficult to calculate than t^ 1

, and we shall not derive it here. Orbach's paper [1961] is now

the classic for this calculation. Equation (22) is

still the basis for the calculation, but the coupling

VSj appearing in the transition probability per unit

time is replaced by

(24)

which is the second term in the standard perturbation expansion. is the sum of the Hamiltonians for the spin and phonon systems, and Ei is the eigenvalue of 3FSj in the initial state. VSj is the same operator as in the resonant-phonon case, but now has the effect of bringing in phonons of other

than the spin-flip energy. For Kramers' salts, t^1

is proportional to the ninth power of the lattice temperature, a result known both experimentally

and by calculation (Orbach [1961] lists several

earlier references). We do not consider non- Kramers' salts, nor the Orbach two-step process

[Orbach, 1961J

In section 5 we consider, among other points,

the effects of the phonon-bath interactions, and

need the damping time characterizing the rate at which the phonons leak out into the bath. In common with earlier work, we simply adopt the

phenomenological notion that the phonon number

decays exponentially with a temperature-independent decay time, related to the time of transit of a phonon across the sample.

In the analysis in the subsequent sections, we use numerical values, in the various relaxation time

formulas, which have been determined empirically

in some cases, or which are representative. The

graphs are given in terms of inverse spin-temperature, which is proportional to magnetization in the

high-temperature approximation:

Tr(e-&*-) Trs (l)

3. Spin Relaxation for Tightly

Coupled Vibrational Modes

In this section, we assume that the resonant modes (designated by d, for direct) are sufficiently

tightly coupled to the remaining vibrational modes

(designated by r, for Raman), that these two sys- tems are effectively a single system with a single

lattice temperature T = (A/3)" 1

. We assume that

a constant-temperature bath is not effective. (In

fig. 1, channels d and e are closed, and c is "wide

open.") In practice, the latter assumption means

that the helium is in fact pumped away after the

lattice has reached its desired initial temperature,

because the spin-lattice times are typically of the

order of 0.01 sec to one second, whereas the lattice- bath times are typically much less than a milli- second. Our interest is in discovering the degree

of deviation from exponential relaxation caused

by the warming-up of the lattice. By eqs (14b, c), if /3d = y32 and /3 r = /33 are equal

initially, which is the usual experimental situation,

then in the limit of small relaxation time r23, they

will remain equal. Equation (14a) then becomes

(1^5)

d(3s _ p-(3s dt where

Ti

—= —+ —= adT+ar T». Tl T d Tr

(25)

(26)

The coefficients ad and ar refer to the direct and

Raman processes, respectively. The T9 law is characteristic of a Kramers' salt. For the second

equation it is convenient to use conservation of

energy, which follows from eq (7) as we have

pointed out:

C^dfc

A/3S2 dt -^=0

kp2 dt

(27)

Here C is the heat capacity of the entire lattice,

given by

C = 234 M-(77©) 3 (28)

in the Debye model for T < 0(O = Debye temperature). N is the number of lattice sites. In terms

6
of the variables T and 5 = k(3s = Ts 1

, eqs (25) and

(27) are

dsldt= (l-sT)(ad + ar Ts

),

and

crds/dt = T3 dTldt,

where cr is defined by

/3

2 CS _ c r

/3

2 C T5

'

By eqs (21) and (28) we have

=

i ns A2 e3 Cr 4 N 234/t2

(25a)

(27a)

(29)

(30)

for S=§. (The subscript r on cr is used to make

the notation consistent with that of the following

sections.) In figures 3 to 11 is displayed the transient be- havior of the inverse spin temperature as given by

eqs (25a) and (27a). The curves in figure 3 are based upon parameter values appropriate to the

spin concentration and splitting used in a para- magnetic crystal which has been studied extensively [Rannestad and Wagner, 1963; Davids and

Wagner, 1964; Baker et al., 1956; Paxman, 1961;

Bray et al., 1962]. This is the crystal K3 Co(CN)6 doped with Fe3+ ion. For Ns/N = 10-3 (Fe/Co = 1.6%), #=2.2, H = 2000 Oe (6.2 GHz), and

0 = 100 °K (0 is not known with certainty [Bray

et al., 1962]), one finds cr ~ 0.1 deg5

. The Raman

relaxation rate coefficient has the value [Rannestad

and Wagner, 1963] ar = 0.0043 sec

-1 deg-9

, and the

direct rate coefficient for H = 2000 Oe has the value

[Davids and Wagner, 1964] a<j ~ 1 sec

-1 deg-1

. Figure 3 is a semilog plot of

5(00) - S(t) s(oo)

versus time, for these parameter values and four

sets of initial temperatures: T(0) = - Ts(0) = 2 °K;

r(0) = -rs(0)=l °K; 7\0) = 2 °K, 7y0) = oo

; and

T(0) = 1 °K, rs(0) = oo. (The condition T(0) = - Ts(0)

is called complete inversion, and the condition

r(0) = oo is called saturation.) The solid curves

give the behavior described by eqs (25a) and (27a);

the dashed curves are straight lines, extrapolated

back from the asymptotic region (where the relaxation is exponential), to show more clearly the deviation of the solid curves from exponential behavior.

It is seen that this deviation is indeed quite small for these parameter values,2 which correspond to low

microwave frequencies, but high spin concentrations. The relaxation is somewhat slower initially

than in the asymptotic region. The reason is that the temperature-dependent relaxation rates in- crease as the lattice temperature slightly rises. From another point of view, one may say that as the spins flip down, the phonon density rises, making

the spin transition rates increase.

For the small direct and Raman relaxation rates assumed in figure 3, a helium bath will keep the

temperatures of the resonant and remaining phonon

modes constant, which undoubtedly is the reason

that the two relaxation rates are found, experimentally [Rannestad and Wagner, 1963; Davids and

Wagner, 1964; Baker et al., 1956; Paxman, 1961;

Bray et al., 1962], to be additive, as we assume in this section. In fact, even if the helium were

effectively removed, boundary scattering of the

phonons is probably sufficiently inelastic that for the rather long spin times used, the lattice modes

are effectively coupled, and would give rise to additive relaxation rates. In figures 4 and 5, we again use ad= 1 sec

-

1

deg -1 and ar — 0.0043 sec

-1 deg-9

, but with cr increased

to 1 deg5 and 10 deg5

, respectively. (Recall [see

eqs (29) and (30)] that an increased value of cr cor- responds to a larger ratio of spin to lattice heat

2 The values of cr used to construct figures 1 and 2 of [Peterson, 1965a] do not cor- respond to the spin concentration, splitting, and Debye temperatures que ted in the

text of that article, due to a numerical error. Figure 3 of the present article is more

nearly representative of the rdCofCNle at 6.2 GHz in the absence of a bath than is figure 2 of [Peterson, 1965a). The cr of this article multiplied by k

5

is the same quantity as c in [Peterson, 1965a].

t (sec) Figure 3. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes tightly coupled. ad =l sec

-1 deg -1

, a r = 0.0043

sec

-1 deg-9

, cr = 0.1 deg5

. The dashed lines are straight lines extrapolated back from

the asymptotic regions.

7
T

1 1 1 1 1 r

t (sec) FIGURE 4. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes tightly coupled. a<*=l sec

-1 deg-1

, ar = 0.0043

sec

-1 deg -9

, cr =l deg5

. The dashed lines are straight lines extrapolated back from

the asymptotic regions.

capacities. An increased value of cr can result from an increase in spin concentration, a higher

Debye temperature, or a larger energy level splitting. The latter can be accomplished by increasing the magnetic field, or sometimes by re-orienting

the crystal in the magnetic field.) Deviations from

exponential recovery are now quite pronounced.

Notice that increased values of cr shorten the decay

times, meaning that the likelihood of observing the

nonexponential behavior is enhanced due to better

competition of these relaxation mechanisms with

any not explicitly included here.

If cr is increased by means of increasing the magnetic field, then the direct rate will also be increased,

going approximately as H4 for Kramers' salts [Scott

and Jeffries, 1962; Orbach, 1961; Kronig, 1939;

Van Vleck, 1940]. We have used ad = 10 sec

-1 deg-1 and ar — 0.0043 sec

-1 deg-9

, with cr = 1 deg5 in figure 6 and cr — 10 deg5 in figure 7, to investigate

this effect quantitatively. The deviation from ex- ponential behavior is now decreased from that in figures 4 and 5. The reason is that the direct

process now tends to "short circuit" the Raman

process. The amount of change of the total relaxation rate, due to the increased temperature of the

lattice, is thereby decreased. Notice also that the

decay times are shortened over those in figures 4

and 5.

t |

1

1 1

1

i | i |

i | r

0 0.1 0.2 0.5 0 4 0 5 0 6 0 7

t (sec) Figure 5. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes tightly coupled. ad = \ sec" 1 deg-1

, ar = 0.0043

sec

-1 deg"', cr = 10 deg5

.

In figures 8—11, we have assumed an increased

Raman rate, using ar = 0.01 and 0.02 sec

-1 deg-9

, with ad — 1 and 10 sec

-1 deg-1 and cr = 1 and 10

deg5

. The nonexponential characteristics are now

more pronounced than in the corresponding figures

4 to 7, and on a still shorter time scale. We remark that figures 3 to 11 are applicable to a much wider range of relaxation rates than is explicitly indicated. That is, if Ti in eq (26) is multiplied by a dimensionless constant /, the only

effect is to multiply the times scales of figures 3 to 11 by/. The shapes of the curves do not change.

To sum up this section, we have exhibited in figures 3 to 11, for a rather large range of parameter

values, the degree to which the spin relaxation

rate increases when the lattice temperature, as- sumed to characterize the entire phonon spectrum,

increases during the relaxation.

4. Spin Relaxation for Uncoupled

Vibrational Modes

In this section, we investigate some of the relaxation characteristics of a spin system coupled to the resonant modes via the direct process, and to the remaining vibrational modes via a Raman T9

8
t (sec)

FIGURE 6. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes tightly coupled. aa= 10 sec"' deg" 1

, ar = 0.0043

sec"1 deg" 1

, c, = 1 deg-5

. The dashed lines are straight lines extrapolated back from

the asymptotic regions.

t (sec) FIGURE 7. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes lightly coupled. aj=10 sec" 1 deg"1

, ar = 0.0043

sec" 1 deg" 1, cr = 10 deg"s

.

process, but for which the resonant modes are

totally uncoupled to the remaing modes.

In eqs (14), we set T231 equal to zero to indicate

the lack of coupling between systems 2 = d and

3 = r. (In fig. 1, channels c, d, and e are closed.)

Equations (14) then become

?here

and

df3x = p d -ps

| Pr-0,

dt Td

dt t,j RS<li

dpr_ /3s-Pr

—77 — Asr, R

dt Tr

Td

1 = adTd , Tr

x = ar T%,

r -Q^L r _

C*&

Ksd -cd L3r

sr Crf3i

(31a)

(31b)

(31c)

(32)

(33)

The heat capacity, Cd, of the resonant modes can

be written

Cd - kp(co)8w = 9/VA2ft8aj/(£2 63). (34)

The first equality in eq (34) results because the resonant modes are at the low-frequency end of the

lattice vibrational spectrum, so that a heat capacity

per mode equal to Boltzmann's constant (the hightemperature expression) can be used. The Debye

formula for the density of modes, p(a>), is used in the second equality. Also, 8oj is a measure of

the bandwidth of modes resonant with the spins,

and N is the number of lattice sites. Setting s — kfis, as before, and using Td and T, as variables in place of fid and /3 r , one can write

eqs (31a, b, c) as

ds/dt = ad(l-sTd) + ar(l -sTr)Tf, (35a)

dTdldt = adcd(l — sTd), (35b)

dTr/dt = arcr( 1 - sTr)n- (35c)

Here ev is the same as that of the preceding section,

defined by eq (29) and given in eq (30) for S= \.

261-405 0-67 —

2
Figure 8. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes tightly coupled. ad =l sec.-' deg~', ar = 0 01 and

0.02 sec"' deg"9

. c r = 1 deg5.

t (sec) FIGURE 9. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes tightly coupled. ad = \ see' dee, '. a r = 0 01 and

0.02 sec" 1 deg-s

, cr = 10 deg5

.

The quantity a is denned by

R

Cd_ T2* 1 d (36)

Combining eqs (21), (34), and (36), one finds

Cd z N 36fi8to

(37)

heat capacity. Because the spin heat capacity is initially typically much larger than that of the resonant modes, the temperature of the latter will rise very high. A qualitative difference in the spin

relaxation occurs, according to whether the spin

system is initially saturated or inverted. For

initial saturation (s0 = 0) eqs (35) give

ds0 dt

(saturation) = ~ (— + —\

To \Td0 Tro/' (38a)

Notice that a is independent of the level separation

A, whereas cr varies as A2

. For the values Ns /N = 10-3

, 6= 100 °K, and a bandwidth 8w equal to a spin resonance linewidth of 8H = 2 Oe (i.e.,

7i8(i) = g(308H, where /30 = Bohr magneton), one

finds that a ~ 105 deg2

. One can predict readily the general features of

the spin relaxation from consideration of the rela- tive heat capacities and relaxation rates, as well as from consideration of eqs (35). Initially the resonant and remaining modes are at the temperature, say T0 . If the direct and Raman rates are comparable at first, the temperature of the resonant mode system will rise much more rapidly than that of the remaining lattice because of its much smaller

(Pso . . . \ — 1 (saturation) = —

\TdO Tro TS-2 ' 1 0 ' roJ

(38b)

where the subscript zeroes refer to the initial time

The initial curvature (d2s 0/dt2

) is typically negative.

For initial complete inversion, eqs (35) give

dso . , 2ds0 (inversion) = -j- (saturation),

dt (38c )

d

2s p dt -y^ (inversion) ~ 2cd

0 ' do (38d)

L0
t (sec) t (sec)

FIGURE 10. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes tightly coupled. a<( = 10 sec -1 deg -1

, ar = 0.01 and

0.02 sec

-1 deg"9

. cr = 1 deg5

.

FIGURE 11. Semi-log plot of transient magnetization after initial saturation and complete inversion.

Resonant and remaining modes tightly coupled. dd = 10 sec

-1 deg-1

. ar = 0.01 and

0.02 sec" 1 deg"9

. cr = 10 deg5

.

In obtaining eq (38d), we have used the fact that

typically

CJL >Y Sl.

1 01 0 we have also assumed that rao is of the order of, or less than tvo- Hence

dt (inversion) >

d

2 s 0 dt' (saturation).

Thus, at initial saturation, the spin relaxation rate begins at the initial direct-plus-Raman rate, but

shortly falls to somewhat less than the Raman rate, when the resonant-phonon energy starts flowing

back into the spin system. This rate then increases

somewhat as the remaining lattice temperature

rises. At initial inversion, the spin relaxation rate begins at twice that for saturation, and very rapidly

increases until saturation is approached. This

process will take place in a time shorter than the

initial value of T<f by a factor of the order of the ratio of To to the maximum resonant-phonon temperature.

This ratio can easily be 10 -4 or less. The subsequent relaxation rate will be considerably slower

than the Raman rate because of the extreme ease

with which the energy of the very hot phonons flow

back into the spin system. The spin system finds

itself almost in a steady-state condition, with nearly constant magnetization. Eventually, the spin re- laxation rate increases, both because the resonant phonons are cooler and the remaining lattice is warming up. The quantitative analysis which

follows now will show that the slowing of the relaxation can be extremely pronounced, and would in fact be readily observable in an experiment for which the

helium bath has been effectively removed, if there were no inelastic scattering of the phonons at the

crystal boundaries. This kind of experiment could

provide a measure of the degree of inelastic boundary scattering. In figure 12, we assume initial spin saturation,

with Td{0) = Tr(0) = 2 °K, ad =l sec" 1 deg-1

, ar = 0.0043 sec" 1 deg"9

, cd =105 deg2

, and cr = 0.1, 1, and 10 deg5

. The three curves may be compared

with the appropriate spin saturation curves of

figures 3 to 5. The first 100 msec of the latter is replotted in figure 12 as dashed curves. The first 20 msec of s(t) and Tait) is plotted on an expanded

scale in the inset of figure 12. The inset shows

clearly the slowing of the relaxation rate after about

3 msec, and the subsequent increase in the rate, particularly for the c r = 10 deg5 curve, due to the

11
100 200 500 400

t (msec) FIGURE 12. Transient magnetization after spin saturation.

Phonon modes uncoupled. 7^(0) = 7V (0) = 2 °K; Oi=l sec" 1 deg -

'; ar = 0.0043 sec"1 deg""; cd = 10s deg2 cr = 0.1, 1, and 10 deg5

. The numbers 0.418, 0.486, and 0.498 are the asymptotic values of s(r) in °K-

'. Dashe

curves are taken from figures 3 to 5, and correspond to tightly coupled phonon modes. Inset: First 20 msec of magnetization and resonant-phonon temperature, on an expanded scale.

warming of the lattice. Overall, however, the

relaxation is not very far from exponential, and an experiment using these parameter values, and

saturation techniques, would likely not distinguish

clearly between effectively coupled, and uncoupled,

phonon modes.

In figure 13, initial spin inversion is assumed, all other conditions being the same as in figure 12. The initial rapid relaxation is plotted in the inset,

which shows both s(t) and Td(t). The near linearity

between s(t) and Tdt) during this initial period is a

reflection of the near conservation of energy be- tween the spin and resonant phonon systems. This

is readily obtained from eqs (35) by setting ar equal

to zero, substituting eq (35b) into (35a), and integrating on time, giving

s{t) « 50 + Td{t)lcd - T0lcd . (39)

Equation (39) can also be used to estimate the maximum resonant phonon temperature, Tm, by setting

Tdit) — [sit)]-1 — Tm- This gives, for an initial partial or complete spin inversion, and cd \so\ To, Tm ~ Cd\s0 \ + T0 + \so\~ 1 ~ cd \s0 \. (40a)

For near-saturation, which may be defined by the

condition \socd — T0 \ <^ c</ 1/2

, Tm « c1 ' 2

. (40b)

Since cd is about 104 to 105 deg2

, the maximum

resonant-phonon temperature can be more than two

orders of magnitude greater after inversion than

after saturation. This readily explains why the

early saturation experiments designed to detect

the hot phonons were unsuccessful, whereas the recent experiments using inversion techniques

have succeeded [Brya and Wagner, 1965; Wagner

and Brya, 1966; Shiren, 1966].

The three solid curves of figure 13 correspond to the left-hand scale. The right-hand scale corres- ponds to the dashed curve, which shows the relaxation behavior in its entirety for cr = 10 deg5

. The

cr — 0.1 and 1 deg5 curves are similar to this, but on

a longer time scale. Of principal interest is the

very flat region in the first 100 msec or so. The

slope there is about 10~4 sec

-1 deg-1

, whereas the

slope due to a purely Raman relaxation in this region would be about sJrro ~ (ToTro)

-1 ~ 1 sec-1 deg" 1

. Figures 14 and 15 are similar to figures 12 and 13,

respectively, Or being increased from 0.0043 to 0.01 sec

-1 deg-9

, all other parameter values being un- changed. The features are qualitatively the same

as in figures 12 and 13. In figure 14 (initial spin

saturation), the resonant phonon temperature does

not rise as high as in figure 12 for the reason that the higher Raman rate coefficient, ar , allows a smaller proportion of the energy to flow into the resonant modes. The "halting" of the spin relaxation in figure 15 is as dramatic as in figure 13,

although the spins are "released" somewhat earlier. Also, comparison of figures 13 and 15 shows that the initial rapid relaxation is quite insensitive to the Raman parameters Or and cr , as expected.

Figure 16 uses the same parameter values as

figure 13, with the exception of cd which is de12
t (msec) FIGURE 13. Transient magnetization after complete spin inversion.

Phonon modes uncoupled. Parameter values same as in figure 12.. Left-hand scale for solid curves; right- hand scale for dashed curve. Inset: First 300 /isec of both s[t) and 7d(t) on an expanded time scale.

creased from 105 to 5 X 104 deg2

. The only gross change is in the initial rapid relaxation from inversion to near-saturation. The initial relaxation now

takes twice as long, and the resonant mode temperature rises only half as high as in figure 13. These

results are in accord with expectations based on

eq (40a). The subsequent slower relaxation is not

quite as slow as in figure 13, the reason being that

a smaller amount of energy is sent initially into the resonant modes.

13
FIGURE 15. Transient magnetization after complete spin inversion.

Phonon modes uncoupled. Parameter values same as in figure 14. Inset: First 300 ji.sec of s(t) and Td(t) on

an expanded time scale.

1

1

1

1

1

1

1 1 r^=f="

S ond Td

Cr

= O.I, 1 . 10 /

1

[ 1 _-J f 1 1 1

i 0.2 0.!

t (msec)

Td CK) /

/

I

I

I J

Figure 16. Transient magnetization after complete spin inversion.

Phonon modes uncoupled. T„(0) = Tr (0) = 2 °K; a d =l sec"' deg-'; a r = 0.0043 sec"' deg" 9

; n= 5x 104 deg 2 cr = 0.1, 1, and 10 deg5

. Left-hand scale for solid curves; right-hand scale for dashed curve. Inset: First half- msec of j(r) and 7Vr) on an expanded scale.

14
Figures 17 and 18 are analogous to figures 12 and

13, respectively, the difference being that in figures 17 and 18 we use Td(0) = Tr(0) = 1 °K rather than

2 °K. In figure 17, both s(t) and Tait) are plotted.

Notice that the ordinate scale is different from that

of figure 12. The most interesting feature of figure 17 is the initial "rapid" relaxation, a feature not so evident in figures 12 and 14. This '"rapid" relaxation is of a different character than that resulting

from an initial spin inversion, and is of course still described by eqs (38a, b). It is rapid only relative

to the subsequent relaxation after about 10 msec,

and in fact is only about half as fast as the initial relaxation of figure 12. After 10 msec, the relaxation proceeds somewhat slower than at the Raman

rate, which is very small because of the low initial temperature, Tr(0)= 1 °K. Figure 18 may be compared with figure 13. The

qualitative aspects are the same. The quantitative differences are the halved initial relaxation

period and doubled maximum Td over that of figure

13, as expected from eq (40a); and the much slower

subsequent relaxation rate due to the lower T0 . Figure 19 is analogous to figure 18, only Cd being

changed, reduced to 5 X 104 deg2 from 105 deg2

. Figure 19 may also be compared with figure 16, for which only the initial temperature is different. In figures 20 and 21, we have changed a-a to 16 sec

-1 deg-1

, and cr to 0.4 deg5

, using ar = 0.0043

sec

-1 deg-9 and c<i=105 deg2 as in figures 12, 13,

17, and 18. These a<* and c,- values would obtain

if the dc magnetic field, applied to a system for which ad =l sec

-1 deg-1 and cr = 0.1 deg5

, were

doubled. The Td ( 0 ) = Tr ( 0 ) = 2 °K curve of figure

20 may be compared with its equivalent in figure

12. An initial "rapid" relaxation now occurs, as a result of the faster direct relaxation rate. We

have also plotted in figure 20, for 7V(0)=2 °K, curves for rd (0)=300 °K, 103 °K, and 104 °K. These values can be considered to be the result of leaving the saturating microwave pulse on for various lengths of time. That is, while the micro- wave pulse is feeding energy into the spin system,

the spin system is feeding energy into the vibrational

modes. The resonant mode temperature will thus

rise significantly if leakage of energy into the bath

or into the remaining vibrational modes is suf- ficiently slow, as we are assuming in this section.

The 7d(0) =2 °K curve would correspond to a short

pulse, less than about 0.1 msec for the conditions

of figure 20, as judged from the Td (t) curve of the

inset, but intense enough to saturate the spin sys- tem. The effect of the longer pulses is to keep

the relaxation from proceeding as far during the

initial "rapid" phase. This effect, the decrease

in amount of relaxation with increase in pulse

length, is reminiscent of the results of Bowers

and Mims [1959], and Bray et al. [1962] in saturation experiments. The duration of the initial de- cays they observed are of the same order as those

in figure 20. However, the amount of decay in the initial relaxation in figure 20 constitutes only

about 1 percent of the total amount [sfafter rapid

decay)/s(final) ~ 0.01], whereas the corresponding

figure in the Bowers and Mims, and Bray et al., experiments apparently can be of the order of 50

percent or more. Bowers and Mims [1959] have

convincingly attributed their initial fast decays to

15
200

t (msec) FIGURE 18. Transient magnetization after complete spin inversion.

Phonon modes uncoupled. Parameter values same as in figure 17. Inset: First 200 ^.sec of s{i) and 7dU) on expanded time scale.

spectral diffusion, that is to the sharing of the

energy between those spins being pumped directly

and those in other parts of their inhomogeneously

broadened spin resonance line. The slowing of the subsequent relaxation due to influx of energy from the resonant phonons is very

evident in figure 20 for the higher Td{Q) values. Notice that the highest value we use (10 4 °K) is still lower than the maximum resonant-phonon temperature which results from initial spin inversion. The

curves of figure 20 can be thought of as indicating

the transition from saturation to inversion.

16
t (msec) FIGURE 20. Transient magnetization after spin saturation.

Phonon modes uncoupled. 7"r (0) = 2 °K; Td(0) = 2 °K, 300 °K. 103 °K, and W °K; a„ = 16 sec" 1 deg" 1

; ar = 0.0043

sec

-1 deg"9

; d= 10s deg2; cr = 0.4 deg5

. Inset; First 5 msec of T^t).

The lower portion of figure 21 shows quantitatively the results of spin inversion from 7^(0) = 7", (0) = 2 °K, and the upper portion shows the results of

spin inversion from 7^(0) = T,(0) = 1 °K, for the same a,i, ar , Cd, and cr values as in figure 20. Notice

that the initial rapid relaxation is now completed

in 10-20 /xsec. As in the preceding section, the curves given in figures 12 to 21 apply to a broader range of relaxation times than explicitly used. That is, if ad and

IT
ar are divided by the same factor f, the time scales

of figures 12 to 21 may simply be multiplied by f to obtain the new curves. In this section we have assumed that the res- onant modes cannot communicate their energy to the remaining modes, and that energy cannot es- cape to a bath. The most significant result of

these assumptions is the pronounced slowing of

the spin relaxation after an initial period of rapid

decay in which spin energy flows into the resonant mode system. This slow rate can be several orders

of magnitude less than the Raman rate appropriate to the conditions used.

5. Spin Relaxation to the Resonant

Phonons Coupled to a Bath

In the preceding two sections we investigated the

transient spin behavior using the assumption that the usual bath had been removed, or could be

ignored. The consequent heating of the resonant modes resulted in two dramatic effects: an initial rapid decay from spin inversion, and a subsequent

very slow decay, slower than the Raman rate. The

initial rapid decay has recently been seen by

Brya and Wagner [1965, 1966]. In this section, we shall examine this latter effect

in a little more detail. We assume that the spins are coupled only to the resonant phonons, which are

in turn coupled to a bath at the constant temperature Tb, (Channels b, c, and e in fig. 1 are closed.)

Spin coupling to the rest of the lattice will be

neglected, because we are here interested chiefly

in the behavior during the first few tens of microseconds. The Raman mechanisms, as we have

seen, have very small influence during this interval. The bath may be helium surrounding the crystal, or it may be the rest of the lattice, if we assume that the temperature of this portion of the lattice does

not change.

Equations (14) become

where

d(3s = pd-ps dt Ta

—J- — K-sd H , at Td Tb

(41a)

(41b)

where Td and RSd are given in eqs (32) and (36), respectively, and T& 1 is the instantaneous relaxation rate characterizing the rate of change of (id = (kTd) -1 due to the interaction between the resonant phonons and bath. In terms of s(t) = k(3s , Tdit), and Tb , eqs (41) are

dsjdt = dd(l —sTd), (42a)

Tph = (Tb/Td)Tb.

Now, the number of thermal phonons at the resonant energy A is given by the Planck factor, which is, in the high temperature approximation,

n = kTdlk.

Thus, by eq (42b), one can also write the rate of

change of Td due to the bath, in the form

dn

dt bath Tph

(43)

dTdldt = adCd(l - sTd) + (Tb - Td)lTph , (42b)

where m is the number of thermal phonons at energy

A at the final equilibrium temperature Tb. The

usual assumption, and the one we adopt, is that the

change of the phonon number due to interaction

with the bath is a simple exponential process. Therefore TPh in eqs (43) and (42b) is taken to be

independent of time.

If the bath is the surrounding helium, TPh is ex- pected to be about Qf/v, where / is the smallest

dimension of the crystal, v is the average phonon

speed, and Q is the number of boundary reflec- tions before escape, which can be from one to about

100, depending upon surface condition and acoustic

mismatch between crystal and helium. If the bath

is the rest of the crystal, Tph would be characterized

principally by inelastic boundary scattering [Faugh- nan and Strandberg, 1961; Scott and Jeffries, 1962].

A lower bound for this would be of the order of

f\v~ 1 /Asec for a single crystal. Phonon-phonon

times at liquid helium temperatures should be

greater than milliseconds. Impurity scattering is elastic. For finite values of TPh, eqs (42) have to be in- tegrated numerically. For Tph = °°, however, the

integration can be performed analytically to give

s(t) = 2Sl + Td(t)lcd, (44a)

Tdit) = [s 2 coth {adCdS2t + (f>) + Si]'1

, (44b)

where

2Sl = s(0)-Td (0)/cd,

52=[5? + Crf

1 ]1 / 2

, *= coth - , (

1 72 So

(

)

0)

)'

; Equations (44a, b) can be manipulated to give also

Cd[s(0) + S2 coth (adCdS-zt)]

The magnetization and resonant-phonon temperaL8
ture thus vary linearly with each other in the high

temperature limit if TPh — x . In figures 22 to 25, we plot the quantity — s(t)Ta{0)

as a function of time for various values of c<f, using

7p/i = x , 10 fxsec. and 5 fxsec, all with 0^=100 sec

-1 deg-1

. The initial temperature 7d(0) is always

taken equal to the bath temperature T&. Notice

that complete inversion corresponds to a value of

unity for the quantity — s(0)Td(0). We consider

in each of figures 22 to 25, three degrees of initial inversion: 1, 0.60, and 0.25. We plot the transient

behavior for only the initial relaxation to nearsaturation; the subsequent relaxation to the bath

is routine. In figure 22, we use Td {0)=Tb = 2 °K and

Cd=: 2Xl04 deg2

. Most apparent is the increasing

rapidity of relaxation with increasing degree of

initial inversion. Notice that the influence of the

bath is quite small for complete initial inversion,

until saturation is nearly reached. This is true even for TPh as small as 5 jxsec. although the initial relaxation takes about 12 /Asec. The main effect of the bath is to flatten the curves somewhat before

saturation is reached. The reason is that the

phonons are being removed, thereby reducing the

amount of stimulated relaxation of the spins.

FIGURE 22. Transient magnetization after three degrees of spin

inversion: 1, 0.60, and 0.25.

TdlO) = T> = 2 °K; a„=100 sec ' deg 1 ; g, = 2x 104 deg2

. The numbers 5, 10, and x are the values of tp h in £isec.

Figure 23 uses the same parameter values as

figure 22, except that Ta (Q)=Tb = l °K. The re- laxation is seen to be somewhat faster in spite of

the slower (halved) initial direct relaxation rate. The reason is that at lower temperatures, the ratio of spin to lattice heat capacity is increased (quadrupled in this case), thereby permitting a higher

phonon temperature to be attained, with its conse- quent greater stimulation of spin relaxation. The

experiments of Byra and Wagner [1965, 1966] were

carried out at about 1.25 °K and 1.5 °K, on the cerium

ion doped into lanthanum magnesium nitrate. The

values of a<* and Cd used in figures 22 and 23 are approximately equal to the values appropriate to their experiments [Wagner and Brya, 1966]. Comparison of their results with figures 22 and 23 shows

that the model used in this section has the features

necessary to at least qualitatively describe the

phonon avalanche. Brya and Wagner (private

communication, preprint) have analyzed their results in some detail along lines similar to those

used here, achieving at least qualitative agreement

with the theory, although there are indications that the resonant-phonon mode bandwidth should be

assumed to increase during the course of the

relaxation.

Figure 23. Transient magnetization after three degrees of spin

inversion: 1, 0.60, and 0t.25. Td [0) = Td = 1 °K; oj=100sec-' deg"'; cd = 2x lO-'deg2

. The numbers 5, 10, and °° are the values of ry h in /xsec.

19
In figure 24, we use Cd — 5 X 104 deg2

, and in fig- ure 25, we use Cd = 105 deg2

, both with 7'd(0)=7&=2

°K. The relaxation is faster and the bath has less influence than in figure 22, as expected. Notice

the changed time scale in figure 25. As mentioned above, we have plotted in figures

22 to 25 only the phonon avalanche portion of the

relaxation. One can readily show from eqs (42) that asymptotically the relaxation is exponential

with a relaxation time given by

r= (a+l)TA0) + terms of order tp„. (45)

Here <x is the "bottleneck factor," a parameter

frequently used [Ruby et al., 1962; Wagner and

Brya, 1966; Faughnan and Strandberg, 1961; Scott and Jeffries, 1962] in characterizing a material for which there is significant phonon heating. It is given by

<r = ctaCdTphTb 1 (46) When <x§>1, the asymptotic relaxation time is approximately proportional to 7V2

, a temperature

dependence characteristically seen in bottlenecked

materials.

0 4 8 12 16 20 24 28

t (/^sec)

FIGURE 24. Transient magnetization after three degrees of spin

inversion: 1, 0.60, and 0.25.

T*{0) = Tb = 2°K; a„= 100 sec" 1 deg" c = 5x 104 deg 2. The numbers 5, 10. and oo are the values of tp i, in fisec.

6. Discussion and Summary

In this article, we have presented the results of

numerical computations of the transient magnetization of a paramagnetic substance, under conditions

in which the lattice can heat up as the magnetic

energy flows into it. Lattice heating can occur

significantly at liquid helium temperatures if the

bath is removed. If the lattice modes are effec- tively coupled to each other, which is likely to be

the case if the time scale of the magnetic relaxation

is of the order of 10~4 sec or larger, we have seen

that deviation from exponential relaxation is not

usually very large, although probably experimentally

observable. The deviation is in the form of an

increasing relaxation rate, due to the increasing

lattice temperature.

If the so-called resonant-phonon modes are completely uncoupled from the remaining phonon modes

(again with the bath removed), two dramatic effects can occur if the spins are initially inverted. First,

there is a rapid decay to near-saturation, due to stimulated emission of resonant phonons, which by

assumption are unable to escape from the crystal.

The energy of these phonons has nowhere to go but

back into the spin system, and does, as the spin

0 2 4 6 8 10 12 14

t (/isec)

FIGURE 25. Transient magnetization after three degrees of spin

inversion: I, 0.60, and 0.25.

T«(0)=Ti, = 2 °K; a,i=100 sec-' deg"'; cd = 105 deg2

. The numbers 5, 10, and °° are the values of tp /, in fisec.

20
system relaxes via a Raman process to the remaining

modes. The result is that for a certain period, the

energy content, and hence the magnetic moment,

of the spin system is nearly stationary, which is the

second effect mentioned above. Eventually the

relaxation rate begins to increase, finally reaching

the Raman rate appropriate to the final lattice temperature. This second effect, the severe suppression of the spin relaxation after the rapid decay,

is expected to be difficult to observe even if the bath

is effectively removed, because of the inevitable

scattering of the phonons from grain boundaries.

This scattering is partly inelastic, and thus partly

short circuits the energy flow of the resonant pho- nons through the spin system. Phonon-phonon

scattering conceivably could also be significant

if the boundary scattering were very greatly reduced.

Quantitative information on these two effects is usually not available. Ultrasonic attenuation meas- urements usually do not distinguish between

phonon-phonon and phonon-boundary scattering.

However, there is little doubt that the direct phononphonon relaxation times are greater than milli- seconds at helium temperatures. Whether the use of a good single crystal with highly polished

surfaces would suppress the inelastic scattering

sufficiently to permit observation of the suppressed

spin relaxation, remains to be discovered.

These considerations suggest, in fact, a possible new way of investigating the amount of inelastic boundary scattering of phonons as a function of

frequency, temperature, crystal perfection, and

surface condition, at microwave frequencies. That

is, the spin signal may be monitored to determine

the degree of suppression of relaxation. The latter, of course, can be calculated, as we have done in section 4, under the assumption of resonant-phonon

decay only to the spin system.

However, it is likely that the results of such ex- periments would only confirm our expectation that boundary scattering is usually strong enough to mask the suppressed spin relaxation. The observations of Brya and Wagner [1965, 1966], the model

for which is discussed in section 5, suggest another

possible way of investigating boundary scattering,

and perhaps also phonon-phonon scattering (which

is temperature dependent [Faughnan and Strandberg, 1961; Herring, 1954]). Measurement of the

spin decay, and hence the hot-phonon decay, after completion of the phonon avalanche, will provide

information about boundary scattering as a function

of the several parameters mentioned above. One

might then attempt to see phonon-phonon effects by using a multi-mode cavity to create two suc- cessive avalanches at two neighboring frequency

bands, and determining whether the decay rate

in one band is influenced by the presence of the

other.

I wish to thank Prof. P. E. Wagner for several

discussions about his experiments, and Dr. J. S.

Wells for his comments on experimental procedures.

I also thank Miss Garney Hardy and Mr. Alfred

Steele for their assistance with the numerical work.

7. References

Baker, J., B. Bleaney. and K. Bowers (1956), Paramagnetic reso- nance in some complex cyanides of the iron group, Proc. Phys.

Soc. (London) B69, 1205. Bowers, K. D.. and W. B. Mims (1959). Paramagnetic relaxation

in nickel fluosilicate. Phys. Rev. 115, 285. Bray, T.. G. Brown, and A. Kiel (1962). Paramagnetic relaxation

of Fe3+ in potassium cobalticyanide, Phys. Rev. 127, 730. Brya. W. J., and P. E. Wagner (1965). Paramagnetic relaxation

to a bottlenecked lattice: development of the phonon avalanche,

Phys. Rev. Letters 14,431.

Casimir, W. B. G. (1938). Note on the conduction of heat in crystals, Physica 5, 495. Casimir. W. B. G., and F. K. DuPre (1938). Note on the thermo- dynamic interpretation of paramagnetic relaxation phenomena,

Physica 5, 507. Davids, D. A., and P. E. Wagner (1964). Magnetic held dependence of paramagnetic relaxation in a Kramers' salt. Phys. Rev.

Letters 12, 141. Faughnan. B. W.. and M. W. P. Strandberg (1961), The role of phonons in paramagnetic relaxation, J. Phys. Chem. Solids 19, 155. Gorter. C. J. (1947). Paramagnetic Relaxation (Elsevier Publishing Co. Inc., New York), pp. 89-90. Hebel, L. C, Jr. (1963). in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic Press, Inc., New York). Vol. 15, pp. 418-420.

Herring, C. (1954), Role of low-energy phonons in thermal con- duction, Phys. Rev. 95, 954. Kronig, R. de L. (1939), On the mechanism of paramagnetic

relaxation, Physica 6, 33. Orbach, R. (1961), Spin-lattice relaxation in rare-earth salts, Proc. Roy. Soc. A264, 458. Paxman, D. (1961), Spin-lattice relaxation time measurements

in dilute potassium ferricyanide, Proc. Phys. Soc. (London)

78, 180. Peterson, R. L. (1965a), Effects of finite lattice heat capacity on

spin-lattice relaxation, Phys. Rev. 137, A1444.

Peterson, R. L. (1965b), Theory of relaxation in a group of weakly

coupled systems, Phys. Rev. 139, A1151. This paper and

[Peterson, 1965a] contain several earlier references giving the derivation and uses of the Pauli equation.

Philippot, J. (1964), Spin-spin relaxation and spin temperatures, Phys. Rev. 133, A471. Rannestad, A., and P. E. Wagner (1963), Paramagnetic relaxation in dilute potassium ferricyanide, Phys. Rev. 131, 1953. Ruby, R. H., H. Benoit, and C. D. Jeffries (1962), Paramagnetic

resonance below 1 °K: spin-lattice relaxation of Cc3+ and Nd3+

in lanthanum magnesium nitrate, Phys. Rev. 127, 51. Scott, P. L., and C. D. Jeffries (1962), Spin-lattice relaxation in some rare-earth salts at helium temperatures; observation

of the phonon bottleneck, Phys. Rev. 127, 32. Shiren, N. S. (1966), Observation of phonons generated in para- magnetic relaxation, Phys. Rev. Letters 17, 958. Van Vleck, J. H. (1940), Paramagnetic relaxation times for ti- tanium and chrome alum, Phys. Rev. 57 , 426. Van Vleck, J. H. (1941), Paramagnetic relaxation and the equilib- rium of lattice oscillators, Phys. Rev. 59, 724. Wagner, P. E., and W. J. Brya (1966), in Physics of Quantum

Electronics, edited by P. L. Kelley, B. Lax, and P. E. Tannenwald (McGraw-Hill, New York), "Stimulated phonon emission from an inverted spin system to a bottlenecked lattice," p. 376. Ziman, J. M. (1960). Electrons and Phonons (Oxford University

Press, London), Chap. XL

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